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I'm currently having a problem with constructibility with ruler and compass as an application of Galois theory.

I want to check the following statement, the proof of which is omitted:

If $\alpha$ and $\beta$ are constructible then so is $\alpha + \beta$. This can be done by showing that $\alpha + \beta \in S(\alpha,|\beta|)\cap S(\beta,|\alpha|)$, where $S(\alpha,|\beta|)$ is the circle with its midpoint at $\alpha$ and radius $\beta$.

I first calculated away by using the equations

$(x - a)^2 + (y - b)^2 = |\beta|^2$ and $(x - c)^2 + (y - d)^2 = |\alpha|^2$ with $\alpha := a + ib$, and $\beta := c + id$.

Five sheets of paper later, I was lost in a jumble of terms and decided to stop right there.

Then I tried axis transformation, but it still came out pretty messy.

Can anyone tell me how to solve this or at least point me to a book in which these constructibility statements are fully calculated?

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    I would never have thought to set your problem up as happening in $\Bbb C$. The treatment I’m most familiar with sets the problem in the Euclidean Plane and talks about the constructibility of distances, once a unit distance is given to you. Do you have a reason for trying to do it all this way? – 2017-02-03
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    We use $\mathbb{C}^{\text{alg}}:=\{\alpha \in \mathbb{C}\;|\;\alpha\; \text{algebraic over}\; \mathbb{Q}\}$ as the algebraic closure of $\mathbb{Q}$. – 2017-02-04
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    Yes, but why? What are your rules for deciding whether an element is constructible? Do you start with $\Bbb Q$ and have a set of rules for getting new elements out of old ones? – 2017-02-04
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    $S(a,r)=\{ z | |z-a|=r\}$ – 2017-03-02

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